Associated Legendre Differential Equation

The associated Legendre differential equation is a generalization of the Legendre differential equation given by


which can be written


(Abramowitz and Stegun 1972; Zwillinger 1997, p. 124). The solutions P_l^m(x) to this equation are called the associated Legendre polynomials (if l is an integer), or associated Legendre functions of the first kind (if l is not an integer). The complete solution is


where Q_l^m(x) is a Legendre function of the second kind.

The associated Legendre differential equation is often written in a form obtained by setting x=costheta. Plugging the identities


into (◇) then gives


See also

Associated Legendre Polynomial, Legendre Differential Equation, Legendre Function of the Second Kind

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Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 332, 1972.Moon, P. and Spencer, D. E. Field Theory for Engineers. New York: Van Nostrand, 1961.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, 1997.

Cite this as:

Weisstein, Eric W. "Associated Legendre Differential Equation." From MathWorld--A Wolfram Web Resource.

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